categories: Derivatives

Given a center-preserving function between intervals, f, is there a
natural way to obtain from it one that's midpoint-preserving? Part of
preserving midpoints is preserving halving: f(hx) = hf(x). We can
rewrite that to convert from a commutativity condition to a fixed-
point condition: f = \x.2f(hx) (the fixed-pont of an operation that
seems to be innate to every graphing calculator). The process needn't
converge, indeed, one will often need to shrink the domain of the
function down to a smaller concentric interval and it can converge to
fixed-points not to our liking (such as \x.x*sin(2pi*log|x|/log2).)
But the closed interval is the final coalgebra not just for the
ordered-wedge functor X v X but for for the ordered-wedge of any
positive number of copies of X. (There's a general theorem that says
that a final coalgebra for X v X is automatically a final coalgebra
for X v X v X v X, indeed, for any such ordered-wedge where the number
of copies of X is a power of two. I haven't been able to find the
proof for a general theorem that specializes to X v X v X.)
Using that the closed interval is the final coalgebra for X v X v X
we can define the thirding map t:I --> I in a manner similar to (and
simpler than) the definition of the halving map. (Yes, the OED lists
"third" as a verb) So we are also looking for solutions to
f = \x.3f(tx).
We obtain a commutative monoid of operators of the form \fx.nf(x/n).
The process of applying a one-generator free monoid of operators
easily generalizes to an arbitrary commutative monoid of operators.
Starting with a center-preserving f between intervals the process
needn't converge even if we restrict the domain to a smaller
concentric interval. But if it does converge to the germ of a
monotonic duality-preserving function then it converges to a germ of a
midpoint-preserving function from the original source interval to the
original target interval. And if these intervals should coincide, it
converges to what in my last posting I defined as an element of the
reals.